diff options
| -rw-r--r-- | BFF.agda | 55 | ||||
| -rw-r--r-- | Bidir.agda | 167 | ||||
| -rw-r--r-- | CheckInsert.agda | 97 | ||||
| -rw-r--r-- | Precond.agda | 44 |
4 files changed, 186 insertions, 177 deletions
@@ -6,9 +6,10 @@ open import Data.Maybe using (Maybe ; just ; nothing ; maybe′) open import Data.List using (List ; [] ; _∷_ ; map ; length) open import Data.Vec using (Vec ; toList ; fromList ; tabulate ; allFin) renaming (lookup to lookupV ; map to mapV ; [] to []V ; _∷_ to _∷V_) open import Function using (id ; _∘_ ; flip) +open import Relation.Binary.Core using (Decidable ; _≡_) open import FinMap -open import CheckInsert +import CheckInsert import FreeTheorems _>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B @@ -17,43 +18,45 @@ _>>=_ = flip (flip maybe′ nothing) fmap : {A B : Set} → (A → B) → Maybe A → Maybe B fmap f = maybe′ (λ a → just (f a)) nothing -module ListBFF where +module ListBFF (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open FreeTheorems.ListList public using (get-type) + open CheckInsert Carrier deq - assoc : {A : Set} {n : ℕ} → EqInst A → List (Fin n) → List A → Maybe (FinMapMaybe n A) - assoc _ [] [] = just empty - assoc eq (i ∷ is) (b ∷ bs) = (assoc eq is bs) >>= (checkInsert eq i b) - assoc _ _ _ = nothing + assoc : {n : ℕ} → List (Fin n) → List Carrier → Maybe (FinMapMaybe n Carrier) + assoc [] [] = just empty + assoc (i ∷ is) (b ∷ bs) = (assoc is bs) >>= (checkInsert i b) + assoc _ _ = nothing - enumerate : {A : Set} → (l : List A) → List (Fin (length l)) + enumerate : (l : List Carrier) → List (Fin (length l)) enumerate l = toList (tabulate id) - denumerate : {A : Set} (l : List A) → Fin (length l) → A + denumerate : (l : List Carrier) → Fin (length l) → Carrier denumerate l = flip lookupV (fromList l) - bff : get-type → ({B : Set} → EqInst B → List B → List B → Maybe (List B)) - bff get eq s v = let s′ = enumerate s - g = fromFunc (denumerate s) - h = assoc eq (get s′) v - h′ = fmap (flip union g) h - in fmap (flip map s′ ∘ flip lookup) h′ + bff : get-type → (List Carrier → List Carrier → Maybe (List Carrier)) + bff get s v = let s′ = enumerate s + g = fromFunc (denumerate s) + h = assoc (get s′) v + h′ = fmap (flip union g) h + in fmap (flip map s′ ∘ flip lookup) h′ -module VecBFF where +module VecBFF (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open FreeTheorems.VecVec public using (get-type) + open CheckInsert Carrier deq - assoc : {A : Set} {n m : ℕ} → EqInst A → Vec (Fin n) m → Vec A m → Maybe (FinMapMaybe n A) - assoc _ []V []V = just empty - assoc eq (i ∷V is) (b ∷V bs) = (assoc eq is bs) >>= (checkInsert eq i b) + assoc : {n m : ℕ} → Vec (Fin n) m → Vec Carrier m → Maybe (FinMapMaybe n Carrier) + assoc []V []V = just empty + assoc (i ∷V is) (b ∷V bs) = (assoc is bs) >>= (checkInsert i b) - enumerate : {A : Set} {n : ℕ} → Vec A n → Vec (Fin n) n + enumerate : {n : ℕ} → Vec Carrier n → Vec (Fin n) n enumerate _ = tabulate id - denumerate : {A : Set} {n : ℕ} → Vec A n → Fin n → A + denumerate : {n : ℕ} → Vec Carrier n → Fin n → Carrier denumerate = flip lookupV - bff : {getlen : ℕ → ℕ} → (get-type getlen) → ({B : Set} {n : ℕ} → EqInst B → Vec B n → Vec B (getlen n) → Maybe (Vec B n)) - bff get eq s v = let s′ = enumerate s - g = fromFunc (denumerate s) - h = assoc eq (get s′) v - h′ = fmap (flip union g) h - in fmap (flip mapV s′ ∘ flip lookupV) h′ + bff : {getlen : ℕ → ℕ} → (get-type getlen) → ({n : ℕ} → Vec Carrier n → Vec Carrier (getlen n) → Maybe (Vec Carrier n)) + bff get s v = let s′ = enumerate s + g = fromFunc (denumerate s) + h = assoc (get s′) v + h′ = fmap (flip union g) h + in fmap (flip mapV s′ ∘ flip lookupV) h′ @@ -1,4 +1,6 @@ -module Bidir where +open import Relation.Binary.Core using (Decidable ; _≡_) + +module Bidir (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) @@ -15,142 +17,143 @@ open import Data.Empty using (⊥-elim) open import Function using (id ; _∘_ ; flip) open import Relation.Nullary using (yes ; no) open import Relation.Nullary.Negation using (contradiction) -open import Relation.Binary.Core using (_≡_ ; refl) +open import Relation.Binary.Core using (refl) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; _≗_ ; trans) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) import FreeTheorems open FreeTheorems.VecVec using (get-type ; free-theorem) open import FinMap -open import CheckInsert +import CheckInsert +open CheckInsert Carrier deq open import BFF using (_>>=_ ; fmap) -open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff) +open BFF.VecBFF Carrier deq using (assoc ; enumerate ; denumerate ; bff) -lemma-1 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : Vec (Fin n) m) → assoc eq is (map f is) ≡ just (restrict f (toList is)) -lemma-1 eq f [] = refl -lemma-1 eq f (i ∷ is′) = begin - assoc eq (i ∷ is′) (map f (i ∷ is′)) +lemma-1 : {m n : ℕ} → (f : Fin n → Carrier) → (is : Vec (Fin n) m) → assoc is (map f is) ≡ just (restrict f (toList is)) +lemma-1 f [] = refl +lemma-1 f (i ∷ is′) = begin + assoc (i ∷ is′) (map f (i ∷ is′)) ≡⟨ refl ⟩ - assoc eq is′ (map f is′) >>= checkInsert eq i (f i) - ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩ - just (restrict f (toList is′)) >>= (checkInsert eq i (f i)) + assoc is′ (map f is′) >>= checkInsert i (f i) + ≡⟨ cong (λ m → m >>= checkInsert i (f i)) (lemma-1 f is′) ⟩ + just (restrict f (toList is′)) >>= (checkInsert i (f i)) ≡⟨ refl ⟩ - checkInsert eq i (f i) (restrict f (toList is′)) - ≡⟨ lemma-checkInsert-restrict eq f i (toList is′) ⟩ + checkInsert i (f i) (restrict f (toList is′)) + ≡⟨ lemma-checkInsert-restrict f i (toList is′) ⟩ just (restrict f (toList (i ∷ is′))) ∎ -lemma-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x -lemma-lookupM-assoc eq i is x xs h p with assoc eq is xs -lemma-lookupM-assoc eq i is x xs h () | nothing -lemma-lookupM-assoc eq i is x xs h p | just h' = apply-checkInsertProof eq i x h' record +lemma-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x +lemma-lookupM-assoc i is x xs h p with assoc is xs +lemma-lookupM-assoc i is x xs h () | nothing +lemma-lookupM-assoc i is x xs h p | just h' = apply-checkInsertProof i x h' record { same = λ lookupM≡justx → begin lookupM i h - ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same eq i x h' lookupM≡justx))) ⟩ + ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-same i x h' lookupM≡justx))) ⟩ lookupM i h' ≡⟨ lookupM≡justx ⟩ just x ∎ ; new = λ lookupM≡nothing → begin lookupM i h - ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new eq i x h' lookupM≡nothing))) ⟩ + ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym p) (lemma-checkInsert-new i x h' lookupM≡nothing))) ⟩ lookupM i (insert i x h') ≡⟨ lemma-lookupM-insert i x h' ⟩ just x ∎ - ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong eq i x h' x' x≢x' lookupM≡justx')) + ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong i x h' x' x≢x' lookupM≡justx')) } -lemma-∉-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing -lemma-∉-lookupM-assoc eq i [] [] h ph i∉is = begin +lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing +lemma-∉-lookupM-assoc i [] [] h ph i∉is = begin lookupM i h ≡⟨ cong (lookupM i) (sym (lemma-from-just ph)) ⟩ lookupM i empty ≡⟨ lemma-lookupM-empty i ⟩ nothing ∎ -lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc eq is' xs' | inspect (assoc eq is') xs' -lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ] -lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof eq i' x' h' record { +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs' +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ] +lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = apply-checkInsertProof i' x' h' record { same = λ lookupM-i'-h'≡just-x' → begin lookupM i h - ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x'))) ⟩ + ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x'))) ⟩ lookupM i h' - ≡⟨ lemma-∉-lookupM-assoc eq i is' xs' h' ph' (i∉is ∘ there) ⟩ + ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩ nothing ∎ ; new = λ lookupM-i'-h'≡nothing → begin lookupM i h - ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-new eq i' x' h' lookupM-i'-h'≡nothing))) ⟩ + ≡⟨ cong (lookupM i) (lemma-from-just (trans (sym ph) (lemma-checkInsert-new i' x' h' lookupM-i'-h'≡nothing))) ⟩ lookupM i (insert i' x' h') ≡⟨ sym (lemma-lookupM-insert-other i i' x' h' (i∉is ∘ here)) ⟩ lookupM i h' - ≡⟨ lemma-∉-lookupM-assoc eq i is' xs' h' ph' (i∉is ∘ there) ⟩ + ≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩ nothing ∎ - ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) + ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) } _in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set _in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is -lemma-assoc-domain : {m n : ℕ} {A : Set} → (eq : EqInst A) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (toList is) in-domain-of h -lemma-assoc-domain eq [] [] h ph = Data.List.All.[] -lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph with assoc eq is' xs' | inspect (assoc eq is') xs' -lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ] -lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof eq i' x' h' record { +lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (toList is) in-domain-of h +lemma-assoc-domain [] [] h ph = Data.List.All.[] +lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc is' xs' | inspect (assoc is') xs' +lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ] +lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] = apply-checkInsertProof i' x' h' record { same = λ lookupM-i'-h'≡just-x' → Data.List.All._∷_ - (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x')) - (lemma-assoc-domain eq is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same eq i' x' h' lookupM-i'-h'≡just-x')) ph))) + (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x')))) lookupM-i'-h'≡just-x')) + (lemma-assoc-domain is' xs' h (trans ph' (trans (sym (lemma-checkInsert-same i' x' h' lookupM-i'-h'≡just-x')) ph))) ; new = λ lookupM-i'-h'≡nothing → Data.List.All._∷_ - (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new eq i' x' h' lookupM-i'-h'≡nothing)))) (lemma-lookupM-insert i' x' h'))) + (x' , (trans (cong (lookupM i') (lemma-from-just (trans (sym ph) (lemma-checkInsert-new i' x' h' lookupM-i'-h'≡nothing)))) (lemma-lookupM-insert i' x' h'))) (Data.List.All.map - (λ {i} p → proj₁ p , lemma-lookupM-checkInsert eq i i' (proj₁ p) x' h' h (proj₂ p) ph) - (lemma-assoc-domain eq is' xs' h' ph')) - ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) + (λ {i} p → proj₁ p , lemma-lookupM-checkInsert i i' (proj₁ p) x' h' h (proj₂ p) ph) + (lemma-assoc-domain is' xs' h' ph')) + ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x'')) } -lemma-map-lookupM-insert : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (h : FinMapMaybe n A) → i ∉ (toList is) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is -lemma-map-lookupM-insert eq i [] x h i∉is ph = refl -lemma-map-lookupM-insert eq i (i' ∷ is') x h i∉is ph = begin +lemma-map-lookupM-insert : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (h : FinMapMaybe n Carrier) → i ∉ (toList is) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is +lemma-map-lookupM-insert i [] x h i∉is ph = refl +lemma-map-lookupM-insert i (i' ∷ is') x h i∉is ph = begin lookupM i' (insert i x h) ∷ map (flip lookupM (insert i x h)) is' ≡⟨ cong (flip _∷_ (map (flip lookupM (insert i x h)) is')) (sym (lemma-lookupM-insert-other i' i x h (i∉is ∘ here ∘ sym))) ⟩ lookupM i' h ∷ map (flip lookupM (insert i x h)) is' - ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert eq i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩ + ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩ lookupM i' h ∷ map (flip lookupM h) is' ∎ -lemma-map-lookupM-assoc : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → (h' : FinMapMaybe n A) → assoc eq is xs ≡ just h' → checkInsert eq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is -lemma-map-lookupM-assoc eq i [] x [] h h' ph' ph = refl -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is')) -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') p -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with eq x x'' -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h .h ph' refl | yes p | (.x , refl) | .(just x) | yes refl = refl -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' () | yes p | (x'' , refl) | .(just x'') | no ¬p -lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p rewrite lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h' ph' ¬p = begin +lemma-map-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → (h' : FinMapMaybe n Carrier) → assoc is xs ≡ just h' → checkInsert i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is +lemma-map-lookupM-assoc i [] x [] h h' ph' ph = refl +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is')) +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') p +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with deq x x'' +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h .h ph' refl | yes p | (.x , refl) | .(just x) | yes refl = refl +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' () | yes p | (x'' , refl) | .(just x'') | no ¬p +lemma-map-lookupM-assoc i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p rewrite lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h' ph' ¬p = begin map (flip lookupM h) (i' ∷ is') ≡⟨ map-cong (λ i'' → cong (lookupM i'') (lemma-from-just (sym ph))) (i' ∷ is') ⟩ map (flip lookupM (insert i x h')) (i' ∷ is') - ≡⟨ lemma-map-lookupM-insert eq i (i' ∷ is') x h' ¬p (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') ⟩ + ≡⟨ lemma-map-lookupM-insert i (i' ∷ is') x h' ¬p (lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h' ph') ⟩ map (flip lookupM h') (i' ∷ is') ∎ -lemma-2 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (is : Vec (Fin n) m) → (v : Vec τ m) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v -lemma-2 eq [] [] h p = refl -lemma-2 eq (i ∷ is) (x ∷ xs) h p with assoc eq is xs | inspect (assoc eq is) xs -lemma-2 eq (i ∷ is) (x ∷ xs) h () | nothing | _ -lemma-2 eq (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin +lemma-2 : {m n : ℕ} → (is : Vec (Fin n) m) → (v : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is v ≡ just h → map (flip lookupM h) is ≡ map just v +lemma-2 [] [] h p = refl +lemma-2 (i ∷ is) (x ∷ xs) h p with assoc is xs | inspect (assoc is) xs +lemma-2 (i ∷ is) (x ∷ xs) h () | nothing | _ +lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin map (flip lookupM h) (i ∷ is) ≡⟨ refl ⟩ lookupM i h ∷ map (flip lookupM h) is - ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc eq i is x xs h (begin - assoc eq (i ∷ is) (x ∷ xs) - ≡⟨ cong (flip _>>=_ (checkInsert eq i x)) ir ⟩ - checkInsert eq i x h' + ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc i is x xs h (begin + assoc (i ∷ is) (x ∷ xs) + ≡⟨ cong (flip _>>=_ (checkInsert i x)) ir ⟩ + checkInsert i x h' ≡⟨ p ⟩ just h ∎) ) ⟩ just x ∷ map (flip lookupM h) is - ≡⟨ cong (_∷_ (just x)) (lemma-map-lookupM-assoc eq i is x xs h h' ir p) ⟩ + ≡⟨ cong (_∷_ (just x)) (lemma-map-lookupM-assoc i is x xs h h' ir p) ⟩ just x ∷ map (flip lookupM h') is - ≡⟨ cong (_∷_ (just x)) (lemma-2 eq is xs h' ir) ⟩ + ≡⟨ cong (_∷_ (just x)) (lemma-2 is xs h' ir) ⟩ just x ∷ map just xs ≡⟨ refl ⟩ map just (x ∷ xs) ∎ -lemma-map-denumerate-enumerate : {m : ℕ} {A : Set} → (as : Vec A m) → map (denumerate as) (enumerate as) ≡ as +lemma-map-denumerate-enumerate : {m : ℕ} → (as : Vec Carrier m) → map (denumerate as) (enumerate as) ≡ as lemma-map-denumerate-enumerate [] = refl lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin map (flip lookupVec (a ∷ as)) (tabulate Fin.suc) @@ -165,16 +168,16 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin ≡⟨ lemma-map-denumerate-enumerate as ⟩ as ∎) -theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} {τ : Set} → (eq : EqInst τ) → (s : Vec τ m) → bff get eq s (get s) ≡ just s -theorem-1 get eq s = begin - bff get eq s (get s) - ≡⟨ cong (bff get eq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ - bff get eq s (get (map (denumerate s) (enumerate s))) - ≡⟨ cong (bff get eq s) (free-theorem get (denumerate s) (enumerate s)) ⟩ - bff get eq s (map (denumerate s) (get (enumerate s))) +theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → bff get s (get s) ≡ just s +theorem-1 get s = begin + bff get s (get s) + ≡⟨ cong (bff get s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩ + bff get s (get (map (denumerate s) (enumerate s))) + ≡⟨ cong (bff get s) (free-theorem get (denumerate s) (enumerate s)) ⟩ + bff get s (map (denumerate s) (get (enumerate s))) ≡⟨ refl ⟩ - fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) (map (denumerate s) (get (enumerate s))))) - ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 eq (denumerate s) (get (enumerate s))) ⟩ + fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc (get (enumerate s)) (map (denumerate s) (get (enumerate s))))) + ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 (denumerate s) (get (enumerate s))) ⟩ fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (flip lookupVec s))) (just (restrict (denumerate s) (toList (get (enumerate s)))))) ≡⟨ refl ⟩ just ((flip map (enumerate s) ∘ flip lookup) (union (restrict (denumerate s) (toList (get (enumerate s)))) (fromFunc (denumerate s)))) @@ -218,16 +221,16 @@ lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin ≡⟨ cong (_∷_ (lookupM i h)) (lemma-union-not-used h h' is' (Data.List.All.tail p)) ⟩ lookupM i h ∷ map (flip lookupM h) is' ∎ -theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {τ : Set} {m : ℕ} → (eq : EqInst τ) → (v : Vec τ (getlen m)) → (s u : Vec τ m) → bff get eq s v ≡ just u → get u ≡ v -theorem-2 get eq v s u p with lemma-fmap-just (assoc eq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) v)) p)) -theorem-2 get eq v s u p | h , ph = begin +theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (v : Vec Carrier (getlen m)) → (s u : Vec Carrier m) → bff get s v ≡ just u → get u ≡ v +theorem-2 get v s u p with lemma-fmap-just (assoc (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc (get (enumerate s)) v)) p)) +theorem-2 get v s u p | h , ph = begin get u ≡⟨ lemma-from-just (begin just (get u) ≡⟨ refl ⟩ fmap get (just u) ≡⟨ cong (fmap get) (sym p) ⟩ - fmap get (bff get eq s v) + fmap get (bff get s v) ≡⟨ cong (fmap get ∘ fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) ph ⟩ fmap get (fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (just h))) ≡⟨ refl ⟩ @@ -238,9 +241,9 @@ theorem-2 get eq v s u p | h , ph = begin map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s)) ≡⟨ lemma-from-map-just (begin map just (map (flip lookup (union h (fromFunc (denumerate s)))) (get (enumerate s))) - ≡⟨ lemma-union-not-used h (fromFunc (denumerate s)) (get (enumerate s)) (lemma-assoc-domain eq (get (enumerate s)) v h ph) ⟩ + ≡⟨ lemma-union-not-used h (fromFunc (denumerate s)) (get (enumerate s)) (lemma-assoc-domain (get (enumerate s)) v h ph) ⟩ map (flip lookupM h) (get (enumerate s)) - ≡⟨ lemma-2 eq (get (enumerate s)) v h ph ⟩ + ≡⟨ lemma-2 (get (enumerate s)) v h ph ⟩ map just v ∎) ⟩ v ∎ diff --git a/CheckInsert.agda b/CheckInsert.agda index 40a57d6..01f1302 100644 --- a/CheckInsert.agda +++ b/CheckInsert.agda @@ -1,4 +1,6 @@ -module CheckInsert where +open import Relation.Binary.Core using (Decidable ; _≡_) + +module CheckInsert (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) @@ -7,86 +9,83 @@ open import Data.Maybe using (Maybe ; nothing ; just) open import Data.List using (List ; [] ; _∷_) open import Relation.Nullary using (Dec ; yes ; no) open import Relation.Nullary.Negation using (contradiction) -open import Relation.Binary.Core using (_≡_ ; refl ; _≢_) +open import Relation.Binary.Core using (refl ; _≢_) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import FinMap -EqInst : Set → Set -EqInst A = (x y : A) → Dec (x ≡ y) - -checkInsert : {A : Set} {n : ℕ} → EqInst A → Fin n → A → FinMapMaybe n A → Maybe (FinMapMaybe n A) -checkInsert eq i b m with lookupM i m -checkInsert eq i b m | just c with eq b c -checkInsert eq i b m | just .b | yes refl = just m -checkInsert eq i b m | just c | no ¬p = nothing -checkInsert eq i b m | nothing = just (insert i b m) +checkInsert : {n : ℕ} → Fin n → Carrier → FinMapMaybe n Carrier → Maybe (FinMapMaybe n Carrier) +checkInsert i b m with lookupM i m +checkInsert i b m | just c with deq b c +checkInsert i b m | just .b | yes refl = just m +checkInsert i b m | just c | no ¬p = nothing +checkInsert i b m | nothing = just (insert i b m) -record checkInsertProof {A : Set} {n : ℕ} (eq : EqInst A) (i : Fin n) (x : A) (m : FinMapMaybe n A) (P : Set) : Set where +record checkInsertProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (P : Set) : Set where field same : lookupM i m ≡ just x → P new : lookupM i m ≡ nothing → P - wrong : (x' : A) → x ≢ x' → lookupM i m ≡ just x' → P + wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → P -apply-checkInsertProof : {A P : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → checkInsertProof eq i x m P → P -apply-checkInsertProof eq i x m rp with lookupM i m | inspect (lookupM i) m -apply-checkInsertProof eq i x m rp | just x' | il with eq x x' -apply-checkInsertProof eq i x m rp | just .x | [ il ] | yes refl = checkInsertProof.same rp il -apply-checkInsertProof eq i x m rp | just x' | [ il ] | no x≢x' = checkInsertProof.wrong rp x' x≢x' il -apply-checkInsertProof eq i x m rp | nothing | [ il ] = checkInsertProof.new rp il +apply-checkInsertProof : {P : Set} {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → checkInsertProof i x m P → P +apply-checkInsertProof i x m rp with lookupM i m | inspect (lookupM i) m +apply-checkInsertProof i x m rp | just x' | il with deq x x' +apply-checkInsertProof i x m rp | just .x | [ il ] | yes refl = checkInsertProof.same rp il +apply-checkInsertProof i x m rp | just x' | [ il ] | no x≢x' = checkInsertProof.wrong rp x' x≢x' il +apply-checkInsertProof i x m rp | nothing | [ il ] = checkInsertProof.new rp il -lemma-checkInsert-same : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → lookupM i m ≡ just x → checkInsert eq i x m ≡ just m -lemma-checkInsert-same eq i x m p with lookupM i m -lemma-checkInsert-same eq i x m refl | .(just x) with eq x x -lemma-checkInsert-same eq i x m refl | .(just x) | yes refl = refl -lemma-checkInsert-same eq i x m refl | .(just x) | no x≢x = contradiction refl x≢x +lemma-checkInsert-same : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ just x → checkInsert i x m ≡ just m +lemma-checkInsert-same i x m p with lookupM i m +lemma-checkInsert-same i x m refl | .(just x) with deq x x +lemma-checkInsert-same i x m refl | .(just x) | yes refl = refl +lemma-checkInsert-same i x m refl | .(just x) | no x≢x = contradiction refl x≢x -lemma-checkInsert-new : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → lookupM i m ≡ nothing → checkInsert eq i x m ≡ just (insert i x m) -lemma-checkInsert-new eq i x m p with lookupM i m -lemma-checkInsert-new eq i x m refl | .nothing = refl +lemma-checkInsert-new : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ nothing → checkInsert i x m ≡ just (insert i x m) +lemma-checkInsert-new i x m p with lookupM i m +lemma-checkInsert-new i x m refl | .nothing = refl -lemma-checkInsert-wrong : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (x : A) → (m : FinMapMaybe n A) → (x' : A) → x ≢ x' → lookupM i m ≡ just x' → checkInsert eq i x m ≡ nothing -lemma-checkInsert-wrong eq i x m x' d p with lookupM i m -lemma-checkInsert-wrong eq i x m x' d refl | .(just x') with eq x x' -lemma-checkInsert-wrong eq i x m x' d refl | .(just x') | yes q = contradiction q d -lemma-checkInsert-wrong eq i x m x' d refl | .(just x') | no ¬q = refl +lemma-checkInsert-wrong : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → checkInsert i x m ≡ nothing +lemma-checkInsert-wrong i x m x' d p with lookupM i m +lemma-checkInsert-wrong i x m x' d refl | .(just x') with deq x x' +lemma-checkInsert-wrong i x m x' d refl | .(just x') | yes q = contradiction q d +lemma-checkInsert-wrong i x m x' d refl | .(just x') | no ¬q = refl -record checkInsertEqualProof {A : Set} {n : ℕ} (eq : EqInst A) (i : Fin n) (x : A) (m : FinMapMaybe n A) (e : Maybe (FinMapMaybe n A)) : Set where +record checkInsertEqualProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (e : Maybe (FinMapMaybe n Carrier)) : Set where field same : lookupM i m ≡ just x → just m ≡ e new : lookupM i m ≡ nothing → just (insert i x m) ≡ e - wrong : (x' : A) → x ≢ x' → lookupM i m ≡ just x' → nothing ≡ e + wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → nothing ≡ e -lift-checkInsertProof : {A : Set} {n : ℕ} {eq : EqInst A} {i : Fin n} {x : A} {m : FinMapMaybe n A} {e : Maybe (FinMapMaybe n A)} → checkInsertEqualProof eq i x m e → checkInsertProof eq i x m (checkInsert eq i x m ≡ e) -lift-checkInsertProof {_} {_} {eq} {i} {x} {m} o = record - { same = λ p → trans (lemma-checkInsert-same eq i x m p) (checkInsertEqualProof.same o p) - ; new = λ p → trans (lemma-checkInsert-new eq i x m p) (checkInsertEqualProof.new o p) - ; wrong = λ x' q p → trans (lemma-checkInsert-wrong eq i x m x' q p) (checkInsertEqualProof.wrong o x' q p) +lift-checkInsertProof : {n : ℕ} {i : Fin n} {x : Carrier} {m : FinMapMaybe n Carrier} {e : Maybe (FinMapMaybe n Carrier)} → checkInsertEqualProof i x m e → checkInsertProof i x m (checkInsert i x m ≡ e) +lift-checkInsertProof {_} {i} {x} {m} o = record + { same = λ p → trans (lemma-checkInsert-same i x m p) (checkInsertEqualProof.same o p) + ; new = λ p → trans (lemma-checkInsert-new i x m p) (checkInsertEqualProof.new o p) + ; wrong = λ x' q p → trans (lemma-checkInsert-wrong i x m x' q p) (checkInsertEqualProof.wrong o x' q p) } -lemma-checkInsert-restrict : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (i : Fin n) → (is : List (Fin n)) → checkInsert eq i (f i) (restrict f is) ≡ just (restrict f (i ∷ is)) -lemma-checkInsert-restrict {τ} eq f i is = apply-checkInsertProof eq i (f i) (restrict f is) (lift-checkInsertProof record +lemma-checkInsert-restrict : {n : ℕ} → (f : Fin n → Carrier) → (i : Fin n) → (is : List (Fin n)) → checkInsert i (f i) (restrict f is) ≡ just (restrict f (i ∷ is)) +lemma-checkInsert-restrict f i is = apply-checkInsertProof i (f i) (restrict f is) (lift-checkInsertProof record { same = λ lookupM≡justx → cong just (lemma-insert-same (restrict f is) i (f i) lookupM≡justx) ; new = λ lookupM≡nothing → refl ; wrong = λ x' x≢x' lookupM≡justx' → contradiction (lemma-lookupM-restrict i f is x' lookupM≡justx') x≢x' }) -lemma-lookupM-checkInsert : {A : Set} {n : ℕ} → (eq : EqInst A) → (i j : Fin n) → (x y : A) → (h h' : FinMapMaybe n A) → lookupM i h ≡ just x → checkInsert eq j y h ≡ just h' → lookupM i h' ≡ just x -lemma-lookupM-checkInsert eq i j x y h h' pl ph' with lookupM j h | inspect (lookupM j) h -lemma-lookupM-checkInsert eq i j x y h .(insert j y h) pl refl | nothing | pl' with i ≟ j -lemma-lookupM-checkInsert eq i .i x y h .(insert i y h) pl refl | nothing | [ pl' ] | yes refl = lemma-just≢nothing (begin just x ≡⟨ sym pl ⟩ lookupM i h ≡⟨ pl' ⟩ (nothing ∎)) -lemma-lookupM-checkInsert eq i j x y h .(insert j y h) pl refl | nothing | pl' | no ¬p = begin +lemma-lookupM-checkInsert : {n : ℕ} → (i j : Fin n) → (x y : Carrier) → (h h' : FinMapMaybe n Carrier) → lookupM i h ≡ just x → checkInsert j y h ≡ just h' → lookupM i h' ≡ just x +lemma-lookupM-checkInsert i j x y h h' pl ph' with lookupM j h | inspect (lookupM j) h +lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' with i ≟ j +lemma-lookupM-checkInsert i .i x y h .(insert i y h) pl refl | nothing | [ pl' ] | yes refl = lemma-just≢nothing (begin just x ≡⟨ sym pl ⟩ lookupM i h ≡⟨ pl' ⟩ (nothing ∎)) +lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' | no ¬p = begin lookupM i (insert j y h) ≡⟨ sym (lemma-lookupM-insert-other i j y h ¬p) ⟩ lookupM i h ≡⟨ pl ⟩ just x ∎ -lemma-lookupM-checkInsert eq i j x y h h' pl ph' | just z | pl' with eq y z -lemma-lookupM-checkInsert eq i j x y h h' pl ph' | just .y | pl' | yes refl = begin +lemma-lookupM-checkInsert i j x y h h' pl ph' | just z | pl' with deq y z +lemma-lookupM-checkInsert i j x y h h' pl ph' | just .y | pl' | yes refl = begin lookupM i h' ≡⟨ cong (lookupM i) (lemma-from-just (sym ph')) ⟩ lookupM i h ≡⟨ pl ⟩ just x ∎ -lemma-lookupM-checkInsert eq i j x y h h' pl () | just z | pl' | no ¬p +lemma-lookupM-checkInsert i j x y h h' pl () | just z | pl' | no ¬p diff --git a/Precond.agda b/Precond.agda index 18d6a85..a6d2d37 100644 --- a/Precond.agda +++ b/Precond.agda @@ -1,4 +1,6 @@ -module Precond where +open import Relation.Binary.Core using (Decidable ; _≡_) + +module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) renaming (zero to nzero ; suc to nsuc) open import Data.Fin using (Fin ; zero ; suc) @@ -8,29 +10,31 @@ open Data.List.Any.Membership-≡ using (_∉_) open import Data.Maybe using (just) open import Data.Product using (∃ ; _,_) open import Function using (flip ; _∘_) -open import Relation.Binary.Core using (_≡_ ; _≢_) +open import Relation.Binary.Core using (_≢_) open import Relation.Binary.PropositionalEquality using (refl ; cong) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import FinMap using (FinMap ; FinMapMaybe ; union ; fromFunc ; empty ; insert) -open import CheckInsert using (EqInst ; checkInsert ; lemma-checkInsert-new) +import CheckInsert +open CheckInsert Carrier deq using (checkInsert ; lemma-checkInsert-new) open import BFF using (fmap ; _>>=_) -open import Bidir using (lemma-∉-lookupM-assoc) +import Bidir +open Bidir Carrier deq using (lemma-∉-lookupM-assoc) -open BFF.VecBFF using (get-type ; assoc ; enumerate ; denumerate ; bff) +open BFF.VecBFF Carrier deq using (get-type ; assoc ; enumerate ; denumerate ; bff) -assoc-enough : {getlen : ℕ → ℕ} (get : get-type getlen) → {B : Set} {m : ℕ} → (eq : EqInst B) → (s : Vec B m) → (v : Vec B (getlen m)) → (h : FinMapMaybe m B) → assoc eq (get (enumerate s)) v ≡ just h → ∃ λ u → bff get eq s v ≡ just u -assoc-enough get {B} {m} eq s v h p = map (flip lookup (union h g)) s′ , (begin - bff get eq s v +assoc-enough : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → (v : Vec Carrier (getlen m)) → (h : FinMapMaybe m Carrier) → assoc (get (enumerate s)) v ≡ just h → ∃ λ u → bff get s v ≡ just u +assoc-enough get {m} s v h p = map (flip lookup (union h g)) s′ , (begin + bff get s v ≡⟨ refl ⟩ - fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (assoc eq (get s′) v)) + fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (assoc (get s′) v)) ≡⟨ cong (fmap (flip map s′ ∘ flip lookup)) (cong (fmap (flip union g)) p) ⟩ fmap (flip map s′ ∘ flip lookup) (fmap (flip union g) (just h)) ≡⟨ refl ⟩ just (map (flip lookup (union h g)) s′) ∎) where s′ : Vec (Fin m) m s′ = enumerate s - g : FinMap m B + g : FinMap m Carrier g = fromFunc (denumerate s) all-different : {A : Set} {n : ℕ} → Vec A n → Set @@ -58,14 +62,14 @@ different-∉ x [] p () different-∉ x (y ∷ ys) p (here px) = p zero (suc zero) (λ ()) px different-∉ x (y ∷ ys) p (there pxs) = different-∉ x ys (different-drop y (x ∷ ys) (different-swap x y ys p)) pxs -different-assoc : {B : Set} {m n : ℕ} → (eq : EqInst B) → (u : Vec (Fin n) m) → (v : Vec B m) → all-different u → ∃ λ h → assoc eq u v ≡ just h -different-assoc eq [] [] p = empty , refl -different-assoc eq (u ∷ us) (v ∷ vs) p with different-assoc eq us vs (λ i j i≢j → p (suc i) (suc j) (i≢j ∘ suc-injective)) -different-assoc eq (u ∷ us) (v ∷ vs) p | h , p' = insert u v h , (begin - assoc eq (u ∷ us) (v ∷ vs) +different-assoc : {m n : ℕ} → (u : Vec (Fin n) m) → (v : Vec Carrier m) → all-different u → ∃ λ h → assoc u v ≡ just h +different-assoc [] [] p = empty , refl +different-assoc (u ∷ us) (v ∷ vs) p with different-assoc us vs (λ i j i≢j → p (suc i) (suc j) (i≢j ∘ suc-injective)) +different-assoc (u ∷ us) (v ∷ vs) p | h , p' = insert u v h , (begin + assoc (u ∷ us) (v ∷ vs) ≡⟨ refl ⟩ - assoc eq us vs >>= checkInsert eq u v - ≡⟨ cong (flip _>>=_ (checkInsert eq u v)) p' ⟩ - checkInsert eq u v h - ≡⟨ lemma-checkInsert-new eq u v h (lemma-∉-lookupM-assoc eq u us vs h p' (different-∉ u us p)) ⟩ - just (insert u v h) ∎)
\ No newline at end of file + assoc us vs >>= checkInsert u v + ≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩ + checkInsert u v h + ≡⟨ lemma-checkInsert-new u v h (lemma-∉-lookupM-assoc u us vs h p' (different-∉ u us p)) ⟩ + just (insert u v h) ∎) |